Solution of a Bi-Objective Purchasing Scheduling Problem with Constrained Funds using Pareto Optimization José Francisco Delgado-Orta1, Laura Cruz-Reyes2, Alejandro Palacios-Espinosa3, and Christian Ayala-Esquivel1 1 Universidad
del Mar, Oaxaca, Mexico
2
Tecnológico Nacional de México, Instituto Tecnológico de Ciudad Madero, Tamaulipas, Mexico 3 Universidad
Autónoma de Baja California Sur, La Paz, Mexico
[email protected],
[email protected],
[email protected],
[email protected]
Abstract. In this paper the Purchasing Scheduling Problem (PSP) with limited funds is presented. PSP is formulated through the optimization of two objectives based on the inventory-supply process: maximization of satisfied demands and minimization of purchasing costs. The problem is solved using two variants of the Ant Colony System algorithm (ACS), designed under Pareto's optimization principle in which elements of multi-objective representation for computing a feasible solution are incorporated to the basic design of ACS. Experimental results reveal that the Pareto approach improves solutions over the ACS in 8%, obtaining an efficiency of 80% solving the set of PSP instances as purchasing plans. This reveals the advantages of developing evolutionary algorithms based on multi-objective approaches, which can be exploited in planning and scheduling systems. Keywords: Purchasing scheduling problem, multi objective optimization, ant colony system algorithm
1
Introduction
The purchase of goods is an essential activity for companies and business. It is the process that involves supply based on searches of items in physical facilities, information of products to check inventory stocks, objects or items in big catalogs and supply of goods on supplier locations. All these activities are periodically executed based on customer demands and the inventory control, associated with the availability
pp. 41–50; rec. 2015-05-11; acc. 2015-07-11
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of economic resources and the storage space in warehouses. In this manner, the Purchasing Scheduling Problem (PSP) (introduced in [1]), establishes a mathematical approach to compute purchasing schedules when demands are variable. Industrial application of PSP is defined as a graph-based problem with several objectives, for example maximization of demand satisfaction, minimization of purchasing costs, maximization of inventory supplies and minimization of supply times. In addition, multi-objective formulation of PSP faces additional constrains such as penalties to influence a schedule with a subset of desired elements, which implies a quality factors in purchasing related with customer preferences [2, 3], critical supply times [4], negotiations in economical lots of orders [5], categorization of products to be purchased [6], and availability of physical space at warehouse facilities [7] when stock must be supplied. For this reason, selection of appropriated goods to be supplied for inventory has become a complex and multi-objective task, whose approach determines the efficiency of a purchasing plan. It is desirable to optimize economical resources in the companies able to produce, distribute and sell their products according to the supply chain.
2
The Purchasing Scheduling Problem
The Purchasing Scheduling Problem (PSP) is defined through a catalog of products like a weighted graph G = (V, E), where 𝑉 = {𝑃 ∪ 𝑆} consists of a set of n products (P) per m suppliers (S). The set E is formed by pairs (p,s), 𝑝 ∈ 𝑃 and 𝑠 ∈ 𝑆. Each pair has a cost cps to purchase a product p from any s supplier. Purchasing process is organized through orders Pk P (or demands), where k represents a decision maker (a purchaser) with a number nk of products to be satisfied with an available fund ak. In these concepts, PSP optimizes two objectives: maximization the amount of satisfied products (for each order Pk) and minimization of purchasing costs (cps) in an inventory cycle. These objectives introduce the field of multi-objective computation.
3
Multiobjective Optimization
Muitivariant and multiobjective nature of real problems present a challenge to development of efficient algorithms. As a consequence, computation of optimal solutions in a multi-objective problem (MOP) is computationally intractable [8] when large-scale instances are solved. As a consequence, optimal solution of MOP is not possible to compute because MOP is represented by a set of objectives in conflict. This is why computation of solutions in a MOP consists of establish the set Pareto front PS = {s1 , s2 ,.., sm} with sm solution vectors of the problem, where feasibility of solutions is given in terms of dominance and efficiency of Pareto. Dominance is defined according to the analysis of objectives in pairs. It establishes that objective s j PS dominates a vector s j ' PS if and only if
s j s j ' , j 1,..., p , with at least one index j for which the inequality is strict
Research in Computing Science 104 (2015)
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Solution of a Bi-Objective Purchasing Scheduling Problem with Constrained Funds using Pareto ...
(denoted by s j s j ' ). Efficiency of Pareto defines a feasible solution sj, for which
z (s j ) z(s j ' ) . It implies that sj is a non-
there does is no other solution s j ' such as
dominated solution (or Pareto optimal). PSP implies the solution of two objectives based on warehouse operations, in which these represent opposite decisions. It defines a multi-objective scene of PSP in terms of a graph-based problem, needed to compute efficient solutions for the related MOP in PSP.
4
PSP Formulation
PSP is formulated through the next data sets: The general inventory catalogue sets: P: is the set of products in an inventory catalog with n products. Pk: is the set of products to be purchased with nk products, Pk P , k=1,2,…,s S: is the set of suppliers in the product catalog with m suppliers. The model uses the next variables: k is the number of orders in each inventory cycle. k = 1,2,…, s cij is the cost to purchase a product i from a supplier j. ak represents the available funds for each order k. xijk is an integer variable {0,1}. It has a one value if a product i is assigned to the supplier j in the order k, zero in otherwise. Objectives of PSP are defined with the f and g coefficients, a normalized objective values in the domain [0,1], where f represents a profit in terms of satisfied demands and g indicates a uniform reference with regard to the assigned cij values for each assigned product. These values are based on the utility principle proposed in [9;10;11], defined through expressions (1) and (2).
max f
1
s
n k 1 k
s
nk
m
k 1
i 1
j 1 ijk
1 min g 1 nk m cij i 1 j 1
x
s
nk
k 1
i 1
,
(1)
. c x j 1 ij ijk
(2)
m
Solution of a multiobjective problem is defined in [12] as a single-objective based on a utility value, following a decomposition strategy. For this reason, objective g is inverted and solved as a maximization objective. As a result, PSP is defined in the general model of expressions (3)-(5): maximize z f g ,
(3)
Subject to:
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Research in Computing Science 104 (2015)
José Francisco Delgado-Orta, Laura Cruz-Reyes, Alejandro Palacios-Espinosa, et al. m
nk
c j 1 i 1
xijk 0,1
ij
xijk ak
k 1,2,..., s
i 1,2,...,n, j 1,2,...,m; k 1,2,...,s
(4)
(5)
The z value of equation (3) has a one-value when all products have been assigned (f is optimal and the dominant objective); in the other hand, a zero-value indicates that g is the dominant objective. Expressions (4)–(5) establishes constrains of available funds in the integer model.
5
The Ant Colony System Algorithm
The Ant Colony System (ACS) algorithm [13] is a well-known method to solve graphbased problems. Construction procedures of solutions in ACS are based on selection of arcs (i,j) of a graph. Ants travel around the roads, leaving an amount of pheromone ij , used to determine the desirability of the roads ij . These parameters are used by artificial ants to generate desirable routes, such as the feedback process of natural ants that looks for the shortest paths between the anthill and the food sources. Evolutive process (iterative) of ACS permits evaporation of pheromone trails to converge towards the most feasible routes, which optimize objectives of the problem. General ACS procedure is presented in Fig. 1.
1
Procedure ACS_Algorithm ()
2
Initialize_parameters ( ij , ij )
3
While(isReached(stopCriteria)) do
4
constructionProcedure( ij , ij )
5
updateOfPheromoneTrailsProceduure( ij )
6 7
End_of_while End Procedure Fig. 1. The AntColonySystem Procedure
The constructionProcedure in ACS_Algorithm builds routes with the desirable nodes in the problem using a transition rule. It defines a basic multi-objective ant colony system algorithm, defined in [1], and based on the multi-objective formulation of PSP. This algorithm creates solutions through of selection of arcs of i products that are purchased to the j suppliers, where selection of the next i-th product is randomly performed in each order Pk. When a product has been selected, the supplier j is chosen using the parameter q0 of equation (6). Research in Computing Science 104 (2015)
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Solution of a Bi-Objective Purchasing Scheduling Problem with Constrained Funds using Pareto ...
*
arg max uNik ij j k f ( pij )
q q0
ij
(6)
oth erwise
When q
